The Hopf algebra of Feynman graphs in QED

Abstract

We report on the Hopf algebraic description of renormalization theory of quantum electrodynamics. The Ward-Takahashi identities are implemented as linear relations on the (commutative) Hopf algebra of Feynman graphs of QED. Compatibility of these relations with the Hopf algebra structure is the mathematical formulation of the physical fact that WT-identities are compatible with renormalization. As a result, the counterterms and the renormalized Feynman amplitudes automatically satisfy the WT-identities, which leads in particular to the well-known identity $Z_1=Z_2$.

13 pages. Latex, uses feynmp. Minor corrections; to appear in LMP

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• An Étude in non-linear Dyson--Schwinger Equations
• From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory
• Renormalization and motivic Galois theory

Cited by in corpus (9)

• Renormalization of gauge fields: A Hopf algebra approach
• Dyson Schwinger Equations: From Hopf algebras to Number Theory
• Recursion and growth estimates in renormalizable quantum field theory
• Feynman integrals and motives
• Diagrammatic Cancellations and the Gauge Dependence of QED
• Non-Linear Algebra and Bogolubov's Recursion
• Supermanifolds from Feynman graphs
• Algebra for quantum fields
• Riemann-Hilbert Problem and Quantum Field Theory: Integrable Renormalization, Dyson-Schwinger Equations